To multiply a binomial and a trinomial, one inserts the terms of each factor into a grid and multiplies corresponding terms. This results in a multiplication with like terms that can then be combined to obtain the final product polynomial. Specifically, the document provides steps to multiply (8x^2 + 16x + 6) and (21x + 2) by first inserting the terms into a grid and multiplying corresponding entries, then combining like terms to get the final product of 168x^3 + 352x^2 + 158x + 12.